Trigonometric Identities

Proving trigonometric identities requires perseverance, a willingness to make mistakes, being able to leave a question and come back to it and a positive mindset.

 

I first wrote out some things we knew.

Solving Trigonometric Equations (a third time)

A review of the compound angle formulas, the double angle formulas and the Pythagorean identity.

All the following questions required to find theta (or x or the angle) between 0 radians and 2pi radians.

First a linear example.

Double Angle Formulas

 

Started by using the compound angle formulas and replaced the second angle so it would be equal to the first. By doing this we were able to generate the double angle formulas for sin2A, cos2A, and tan2A. Here is some work from various groups.

 

Compound Trigonometric Formulas

 

Mathematical Processes

As a class we explored what the seven mathematical processes meant to us. Here was our brainstorm.

 

REFLECTING

Creating the Trigonometric Equalities Between Different Angles

This post shows the relationship between theta, pi - theta, pi + theta, 2pi - theta, pi/2 - theta,
pi/2 + theta, 3pi/2 - theta, 3pi/2 + theta.

Solving Trigonometric Equations ( a second time)

Revisiting how to solve a trigonometric equation.

Some examples where we get a linear equation to solve:
This won starts with a reciprocal trigonometric function



 

Solving Exponential Equations using Logarithms

In this post we look at exponential equations where we cannot solve them by getting a common base on both sides. We accomplish this by taking the log of both sides and then applying the logarithm laws to both sides to allow us to solve for the unknown.

An example:



Solving Logarithmic Equations

This post pertains to solving Logarithmic equations using the Log laws.
An easy example:

 

Solving Exponential Equations by Getting the Same Base

This post is about solving exponential equations by getting the same base on both sides. Since the bases are the same the exponents must be equal. Some examples from class:

 

 

Solving Rational Equations

Solving rational equations is equivalent to finding the y-intercepts of a rational equation.
To make sure you do not lose any roots the best way to do this is to put everything on one side and then solve. Here are examples we worked on in class:

 





 

Solving Polynomial Equations

Solving polynomial equations is equivalent to finding the y-intercepts of a polynomial equation.
If y = x^3-4x^2+8x-32 then to solve 0 = x^3-4x^2+8x-32 is the same as finding the x values that make y = 0. Here are two examples multiple times:

Logistic Functions

Combining Functions (Adding, Subtracting, Multiplying, Dividing)

 

Solving Trigonometric Equations

After the ferris wheel we formalized solving trigonometric equations. Here is the sequence of questions we worked on.

A trig function equals some number.

Ferris Wheels

Students were asked to write any questions they had on this video of a ferris wheel. As a class we settled on " where does the red bucket finish at the end of the ride?" Then we decided what info we needed. We collected some data from the video. pi on 4 radians every 5 seconds. Students then solved the question for a three minute ride.